Problems with Zero - Numberphile
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- čas přidán 20. 05. 2024
- Dividing by zero, zero divided by zero and zero to the power of zero - all pose problems!
More links & stuff in full description below ↓↓↓
This video features Matt Parker and James Grime - / standupmaths and / jamesgrime
NUMBERPHILE
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Matt is very smart for a guy who writes infinity as double zeroes instead of a laying eight
What if he writes eight the same way?
And X as two C having each other's back
@@marzi_kat It's treason, then.
@@rorschak47
Well that’s cause it’s a cursive x
@JZ's Best Friend They would put him in a room with a quadratic floor and call it the Parker Square.
"BUT, if I am naughty..." Oh baby, talk nerdy to me~
LOL
+James Lynn Savage
+James Lynn
Rounding to the first decimal place also counts as naughty. Computing ∫e^x dx makes one horny.
+James Lynn oh my lord
pearl you're smart, what's the real answer here to 1/0
"glorified adding" is the best description of multiplying ever
would that mean exponents are extra glorified adding?
@@BasicEndjo glorified multiplication
@@shnob4916 yeah and multiplication is glorified adding. then exponent is *extra* glorified adding
An article in the Onion from 1907 reported that a record breaking number of American children are staying in school beyond third grade. They are learning advanced skills such as multiplication, which we are told, is a powerful form of adding, resulting in numbers so large that three or even sometimes four figures are required to write them.
This is what I told my 2 grader. “Fast addition”
My 7th grade algebra teacher would only whisper of dividing by zero because it would “upset the calculator gods”. He was one of my favorite teachers ever.
Great anecdote
My math teacher said he could taste numbers
@@malteepmeier Ah, Synesthesia!
@@malteepmeier mine snorted cocaine in class and offered some to us.
@@cones914 Disgusting! Where is that teacher located? So I can avoid it
"Only a nerd would tell you differently."
*cuts to Parker* - Sooo, first of all [...]
Even better "... and that's when you cut to Matt telling them differently."
i clicked the [...] in ur comment wtf wrong with me lol
"And then we cut to a nerd telling you differently."
exploshi I clicked the [...] on yours😂
thatsthejoke.jpg
I suggest we define 1/0 = blue
Jonathan Tanner
Look at me
Jonathan Tanner make a petition
1÷0=blue. The newest axiom in mathematics.
x/0 = blue(x+1)
@@bsm239 1÷0=Blue×1+1=Blue×2=2Blue
"The problem is it's a dangerous number and a lot of things can go horribly wrong with 0"
"Mom I got 0 in maths"
*UH OH*
Lol
if it was art, it would’ve been better
@@boncoderz1430 you mean nice zero?
@@tenplusten1116 Especially if you're from Austria...
@@leyyesuh oh…
A video featuring both Matt and James is such a lovely treat. They are infinitely different.
i see what you did there hahah
Gosh you can’t say that
1/0 = blue the secret is out
Lol.
Oh deah, it's a poblem
He used a 1/0 Sharpie.
Shhh, You're not supposed to tell anybody
Are you talking to Blue the Dog about the secret?
*Santa:* You don't get presents this year because you were naughty
*Mathematician:* What! Why?
*Santa:* You used infinity as an answer
in high school I was doing a problem on the blackboard in an algebra class and I was finishing it fast so I was writing both sides of the equation (my way of doing it) and the teacher saw it and yelled "algebraic sacrilege!!!!" and that scared me lol and everyone else in the classroom. I swear that we almost hear thunders falling on us. Sufficient to say that I never had the opportunity to explain to him that I was still writing my answer, so I just completed the answer as a "correction".
@@marilynman You're supposed to write both sides?
@@circuit10 I meant that when you start writing it, you begin with the right side, you finish writing that side and then move on to the right side. I was doing both sides at the same time.
X/0 = santa
@Eero Naughty boy. Now you cannot wish to Santa anymore. Even finding out the meaning of life and solutions to infinity.
"Divide"
-No
"GlOriFieD SuBsTrAcTioN"
- *YES*
Substraction?
That is like roblox ad...
Thank me for i was the 69th to like
Jk
@@niccolopaganini1782 hello paganini
0/0 = calculus
12:49 "we could make it anything we want it to be depending on the angle we come at it from"
sound life advice right there
For the "why does it return Error in a computer" question, the division assembly instructions (at least for x86) are designed to generate an interrupt when the divisor is zero. In other words, they are told to error out.
What would happen if they werent?
@@Xnoob545 Presumably it would attempt that forever. It'd never find its result, and the part that tells it to stop has been chopped off, so it'll just never stop
@@Xnoob545 setcomputeronfire.exe would initiate and well... You get the idea.
@@y-ax2-bx-c5 its like a minecraft world made out of sand
Just falls fprever amd uses exponentaially more power
@@Xnoob545 cpu would hang at 100% usage trying to compute the result of what cant be computed, until you restarted it. therefore safety instruction/lock was added to prevent such.
Plot twist; the entire Numberphile series is a promotion for Sharpie.
vihart???
Andrew Jones - And brown paper.
I can smell the sharpies.
And the brown paper
DUN DUN DUN
Now, I had always been taught that X/0 was "undefined", while 0/0 was "indeterminate". The logic behind this is that the denominator (or "divisor") should always be able to be made equal to the numerator, by multiplication with some factor.
So, for example, 1/2 = .5, thus 2 can be made equal to 1 by multiplication with.5. However, in the case of X/0, there is no factor that can make 0 = X, since 0 times ANYthing is always 0. So, there is no correct answer, therefore, the problem is "undefned".
On the other hand, in the case of 0/0, literally ANY factor will make 0 equal to itself, so there is no INcorrect answer. Thus, in essence, any value is equal to any OTHER value, which is impossible. Therefore, the problem is called "indeterminate", since one cannot determine what value best solves the problem.
@Vikas Bhardwaj what if we define 0/0 as equal to 0
I know you said this is what you were taught, but it bears mentioning that this is just incorrect. There is no such a thing as "indeterminate" in mathematics, and people need to stop using this word forever. 0/0 does not exist. Period. That is all there is to it. And there is a very simple reason it does not, but it just has to do with what division itself is. Division is just multiplication: multiplication by the reciprocal, to be exact. 0 has no reciprocal. So one cannot divide by 0.
@@angelmendez-rivera351 because in calculus, 0 is not exactly 0, 0 can be 0.0002 or -0.00001, numbers are not exactly their values. That’s why there is indeterminate
@deaf I fail to see how those points connect. 0*x = 0 for all values of x is a true statement. I don't see how this implies that 0/0 = 1 any more than it does any arbitrary number.
@@angelmendez-rivera351 Yes, in terms of numerical value, indeterminate forms are considered undefined. But they are very useful in calculus because of how they affect limits. (f(x+h) - f(x))/h = 0/0 when h=0, so it's undefined. But the limit as h approaches 0 is very much defined (when f(x) is continuous), and is in fact the definition of the derivative. If 0/0 is just undefined, derivatives don't exist, and calculus doesn't work. That's why we have indeterminate forms, at least when working with limits
There's a video around of an old mechanical calculator which gets stuck in a loop when trying to divide by zero, and the operator has to press the abort button to stop it running.
Nothing bad happens - it just keeps subtracting zero and counting how many times it subtracts zero and it never finishes.
"We have to slide it in, from both directions." - Phwoar........
stop. lets keep it PG
+Sarah Cartwright Hi Sarah!
+John Yyc zero ain't PG mate. Zero is a dirty boy.
+Sarah Cartwright "So you say, 'Well, if it doesn't matter which side we're coming in from... surely we can just call it one.'"
+John Yyc Even PG movies are allowed brief adult moments.
I love these numberphile videos because you can litterally notice how they get high on math as the video goes 😂
i see much enthusiasm :D I suggest "a hole in a hole in a hole" :D
Math makes me horny
It's the only way to fly.
That's what happens when you inhale sharpies.
*Meth
There's some great footage on CZcams of mechanical calculators, oldschool ones, dividing by zero. No programmed-in "Math Error" there, the things just spin forever making a racket, they're probably subtracting zero over and over but maybe some of them are failing in a more clever way.
Error; task completed successfully 😂
A decade later and still a fantastic video!
Ten, meaning one, zero. Coincidence? I think not!
I divided 1/0 in my calculator and now it runs Super Mario 64.
if I divide by 0 on my smartphone, can my smartphone calculate a rocket launch?
@@farisakmal2722 Probably
I divided zero on my table and now it can shoot lasers and fly
ok............
I inputted 1/0 into a sideways 8 and now every time I draw an 8, it Runs fortnite
Everybody in this comment section is a math genius....
welcome to the *light side* of CZcams, buddy.
Literally, it was such a culture shock for me when i 1st came here XD!
Chris Griffin Hah! MY EYES! Yes... YT is changing :)
Except from me :(
'Maths genius?'
Ohnhon this is stuff which, if you ever paid attention in school, should be logical. I don't understand why people believe that worked out concepts are so hard. You just gotta puzzle over it until you understand.
The word "everybody" is actually conjugated in the singular, as in "everybody was Kung Fu fighting", and not "everybody were".
Genii is acceptable, however.
6:10 Yes, computers are taught not to divide by 0. The reason is because bitwise math operations are only add and subtract. Multiplication is just repeated adding, while division is repeated subtracting. If you divide by 0, you are telling the computer to subtract 0 from the original until the value of the original is
≤ not
Happy New year sir
But Javascript gives back Infinity..
@@jathebest2835 That doesn't matter since the sign of the quotient is determined by the dividend. So are you going to get it as infinity or negative infinity? Since the answer is undefined anyways, is there a point in computing it?
@@mdsharfuddinmd5710 hi you too
11:23 Even the painting is interested in mathematics.
lOL
Don’t expose them to sunlight, don’t let them eat after midnight, don’t get them wet, and never divide by zero
I understood that reference!
But it’s always after midnight...
Do not touch the operational end of The Device. Do not submerge The Device in liquid, even partially. Most importantly, under no circumstances should you divide The Device by zero.
AND NEVER EAT PEARS!
… And those are the rules for math gremlins.
1÷0= infinity
2÷0= Double infinity
There I Fixed it
@Interesting Numbers Wait a minute, then you can multiply both sides by 0 and get 1=2. I think the Illuminati must be at work here. -lol
@@coulombicdistortion1814 Here's the thing about multiplying by zero: anything multiplied by zero is zero. So 0(1/0=2/0)
(1*0)/(0*0)=(2*0)/(0*0)
0/0=0/0
Travis Ryno I have been stuck on this since last evening. My 13 yr old told me exactly this, and then used Banach-Tarski model to say that 1+1=1 is mathematically possible. I don’t know right now whether to believe his hypothesis or continue to say x/0 is undefined.
@@annoyinglyfast5972 He is saying that the logic is wrong
@Travis Ryno therefore 0/0=0
Multiplication and division are the next iteration of addition and subtraction. The iterations beyond those are exponents and roots. When you get beyond that, it gets really hairy. Layered exponents (also known as "towers") and roots (just the reciprocal of towers) are as far as most people dare to go. But you can technically go as far as you want, and Knuth invented a special notation to explain the weird realm beyond layered exponents/roots. It was used to create one of the largest numbers ever conceived, Graham's number.
2:25Mathematicians definition if “naughty” and “evil” - “Ooh, what if I said 1/0 = infinity? Ooooooh”
"We can no more say that 1 divided by 0 is equal to blue"
I lost it
"we can no more say that *than* [that] one divided by zero is equal to blue"
you missed a "than"
@@meta04 ?
÷0 looks like a screaming person
Yep
Fitting
It's screaming at you for trying to destroy the universe.
Thats the reason you cant devide by zero.
But you can devide zero.
Or like a key
The way he writes Infinite Haunts me 3:07
5:50 The look you get, when you're on a date with a mathematician
Joke's on you, I divided by zero and got an answer. I put 1/0 in my calculator, and got "Error". So, 1/0=Error.
Jude Pelaez makes sense lol
basically, you get something, what programmer wrote. And of course he lies.
Jude Pelaez 0 divided by 0=error
Jude Pelaez So 1/0 is the guy from Zelda 2? Brilliant!
Jude Pelaez but what about 0 divided with 'error'?
When i was a child. I thought 0 and -0 were different numbers, and i kept counting wrong when going from positive to negative or negative to positive
In IEEE 754 format they *are* different numbers. They behave very much alike, though.
What's bigger, zero minus zero, or zero minus negative zero? Lol just kidding.
@@p.as.in.pterodactyl1024 Who is bigger, Mr. Bigger or Mr. Bigger's baby? The baby, because he's a little bigger.
Original commenter: You were born with what is called "Ones' complement"! I guess you were upgraded to "Twos' complement" since then.
@@user-yv1qs7sy9d What is that?
6:25 totally agree with that. I know that when I do a really intense calculation on Desmos, the calculator displays to me a message saying "definitions are nested too deeply"
Also, it can’t be infinity, because even if you subtract it an infinite amount of times your still going to have the number you started with.
But all division does is count the subtractions that took place to reach the number. Therefore, it isn't infinity or the number you started with. It's 0.
20 / 4 = 5 (Five Subtractions)
20 - 0 = 20. No subtraction took place.
20 / 0 = 0 (Zero Subtractions)
@@Vespyr_ No, that is horribly incorrect. Firstly, that is not how division actually works: division is not repated subtraction, and multiplication is not repeated addition. Secondly, even if division did work that way, your answer is still wrong, becaue 20/0 would be equal, by your definition, to the number of times you have to subtract 0 from 20 to achieve 0. The problem is that, even if you subtract 0 an infinite amount of times from 20, you still do not achieve 0. The answer is not 0, nor is it an infinite number. It is just impossible to achieve 0 via such repeated subtractions, hence 20/0 is undefined.
Nevermind this, because as I explained firstly, division is not repeated subtraction. The reason division by 0 is problematic is because, in order for division by a quantity A to be possible, you need to have the following property: if A·x = A·y, then x = y. This does not occur with 0. 0·1 = 0·(1 + 1), but 1 = 1 + 1 is false, in general. So division by 0 is hopeless.
The answer is super existence, a level above every number
@@angelmendez-rivera351 I think you missed the point of glorified subtraction but that idea does work, 28 divided by 4 is just 28 minus 4 over and over till its 0, which is when it's been subracted 7 times
@@thefloormat3297 No, dude, I literally addressed it within my first sentence. Maybe you do not know how to read. Also, I already explained how subtraction does not work. You cannot subtract 0 over and over from 20 until you get 0. It is impossible.
7:44
"We're gonna slide it in and, in fact, we're gonna have to do it from both directions"
O___0
Liam Dienemann
“more than a numberphile”
An Infinityphile XDXDXDXD
There's a reason why infinity is drawn as two circles.
Nice
An accountant, an engineer, and a mathematician are asked how much is 1 + 1:
Mathematician: "1 + 1 is 2 and I can prove it"
Engineer: "Well, 1 + 1 is anything between 1.8 & 2.1"
Accountant: "It depends. How much do you want 1 + 1 to equal?"
Democrat: 1 x 0 = transgender
Republican: 1 + 1 = homophobe
Quantum physicist: i don't know until i actually calculate it
Surrealist: yes
Communist: 1/2 for me, 1/2 for you, and 1 whole for the state
It depends whether we sell or buy.
How to make a million dollours on paper with 50
Numberphile: Imma head out
Software developer here: We don't try to solve it by any method...
we throw error for situations like this 😂
@@ForeverStill_Fan1 start with C or C++
Understand the basics of programming...
Don't directly jump to python... Python is a high level language doesn't help too much for building logic...
For frameworks- depends on your interest
If I understand my computer science right, computers' physical arithmetic processing units throw errors when they're ordered to divide by zero, which would cause horrible breakage. In practice, though, the command to divide by zero is intercepted by stuff like the operating system long before it actually manages to reach the hardware.
The way he smiles when he brings in the complex numbers...
And that all the while glossing over X^X for negative X looking really strange (it's jumping all over the complex plane and is basically discontinous everywhere). That is not a function for which you want to find a limit. The complex version must be just as bizarre.
"Maybe this line goes all the way around and wraps around the entire universe and things come back up here"
I'm having vertigo
Not surprising if you know the history of the projective plane...
@@digitig *flashbacks
I've always interpreted it as being positive and negative infinity and every possible number in-between.
Not sure if that's valid though.
@@StevenAyre1 yes, 1/0 is every number ever to be thought of at the same time...
I recently learned what the actual name for 0/0 is in Calculus. It's called an indeterminant, because it can give any answer. If we want to solve it, we need to know the function that created the 0/0, as they show. Then we take the derivative of the top and the bottom (separately), and try to divide again. We repeat until we don't get a 0/0
L'Hopital's!
That’s not quite accurate. An indeterminate form like 0/0 is entirely meaningless on its own and fundamentally can not be “solved” unless you’re talking about in terms of a limit.
Furthermore, 0/0 isn’t the only such form, so your use of L’hopital often isn’t applicable.
@@cpotisch All the other indeterminate forms can be turned into a 0/0 or inf/inf form, in which L'H can then be applicable
@@gamerdio2503 e^x-x as x goes to infinity?
@@cpotisch Fair enough. Most of the time, the indeterminate form can be converted into a form usable with L'H. Although, you can just use the fact that e^x grows faster than x to get a quick answer
This is my favorite video on this channel. Makes me chuckle every time.
3:05 Noooooooooo!!!
Draw infinity as a continuous loop not two circles!!!
It's ugly
@Al Gee Writing it as a continuous loop has a more comfortable flow than two separate circles
two circles are not a lemniscate!
Yeah, the entire point of the symbol is being a literal neverending loop
haha infinity go oo
Most CPU's have a specific return code for "Divide by Zero Error", meaning it doesn't attempt to calculate as the error is handled at the CPU level.
Number_055 return 0;
I think it's implemented at OS level. And older operating system just tried to subtract 0 from the number forever, forcing you to turn off the power and turn it on again.
@@DanCojocaru2000 If you actually think that applications call the OS to perform calculations then you haven't got a clue what an operating system actually does and doesn't do. Division is not a system task, it can be performed by any application that is directly using the processor (CPU) at user level. Actually, most CPU's have division build right into them...sometimes incorrectly, check the Pentium bug (for floating point division).
Specifically, it's a hardware trap (ie. hardware exception) generated by the CPU at the time of division, at least on x86 / x86-64.
Edit: I made a mistake in my original post, and I apologize. A divide-by-zero will return a "not a number" (NaN) result for a floating-point division. I don't know off the top of my head what result an integer division returns - this is something I should either look up or simply test - but the divide-by-zero register is still set, which can be queried to determine if an exception should be thrown. Floating point values may contain infinity and negative infinity as actual values, and if you want, you can treat a divide-by-zero as infinity, and I have in fact seen API's that do this, but generally speaking, you don't want a divide-by-zero to ever be a valid operation.
Original post:
@@majormalfunction0071 Intel CPU's will happily divide by zero and return either infinity or negative infinity, depending on the sign of the operation - they also differentiate between zero and negative zero; the sign is simply a bit in the return value. They will, however, as Number_055 noted, also set a "divide by zero" register notifying the application requesting the operation that a division by zero occurred, which the application may then treat any way it likes, including treating the operation as an exception and possibly crashing itself. I wholly agree that "infinity" is not a valid return value for a divide-by-zero, but the IEEE standards committee had to settle on something that would work from a technical standpoint.
I still remember that day when I was in the middle school. Our math teacher, let us use 1 divide some positive numbers smaller and smaller, than we found the results bigger and bigger. Then we use negative numbers bigger and bigger, and the results were smaller and smaller. On that day all of us remembered we cannot use some numbers simply to divide 0.
"anything divided by zero is counting to infinity."
Vsauce: *how to count pass infinity*
Division by non-existence?
It makes no sense.
When you asked the old Sinclair calculator from the 70s to divide by zero, it actually tried! It would give you multiple answers one after the other until eventually it spat the dummy, showed all the decimal points and locked the screen.
spat the dummy? that isn't a phrase
Ben P it is in Straya
0 is my favorite number because it has no value, just like me.
I read this and I laughed. Thank you very much for that laugh.
:)
ᎢᎻᎬᎠᎾᎠᎾ ᎬNᎢᎻᏌᏚᏆᎪᏚᎢ jokes on you, our numbers don't work without a concept of zero
ᎢᎻᎬᎠᎾᎠᎾ ᎬNᎢᎻᏌᏚᏆᎪᏚᎢ CRAWLING IN MY CRAWL
Bad day?
Ooh ..hope that was limited to being a joke.
A number divided by zero lacks an equation. The person asking
"What does 5 ÷ 0 = ?"
Is like him asking,
"What does 5"?
See? When zero follows the division symbol then no equation takes place. We've left the realm of mathematics and returned back to language. I'm surprised that I've never seen this explanation offered before. It is however the only one that explains the question.
When we attempt to divide by 0 we're no longer dividing. Its like any attempt to divide with zero automatically erases the whole procedure.
This is true for processors and the DIV instruction, when you try to divide by 0 there it fires an interrupt that basically means "Result is undefined". (It basically checks if there's a 0 anywhere in the 'equation', and if there is, it doesn't calculate anything) I'm not sure if that's true for all processors, but the ones I have experience with use the interrupt method.
I find I don't understand what you're saying here, which concerns me as you say that it's the only phrasing that successfully answers the question. How is it that you're able to say that "divided by zero equals" is equivalent to saying nothing at all? If math has proven capable of making proper use out of numbers that don't even exist (square root of negative one, an imaginary number), then why does this simple utterance disappear when nothing else does?
Dude who makes the infinity sign like that 3:09
Who cares how it looks only what it means matter
The computer is actually taught to not divide by zero. There are many situations in software where dividing by zero is caught and protected against. My brother used to work in a hardware store and he had a computer that gave a 'divided by 0' blue screen. According to the story, he laughed insanely laud at that blue screen. Usually that doesn't happen but the computer had a defect RAM which fed corrupted data into the processor as it fetched the information to execute the micro programs. The processor actually had a build in protection to prevent dividing by zero, it stopped the operation and 'breached' away from its micro instruction to the error handling of windows which on its term showed the blue screen.
In short, the computer doesn't even attempt to divide by zero. If you were to try and do it it would probably try to apply a form of implemented long devision which would obviously fail and I have no clue what it would return.
Robert sorry CZcams isn't letting me post my own comments one thing I would note for the people at number phile is it's as easy as defining 0*y=0 y in the complex plane but not =0 so 0 divided by 0 makes no sense since you can turn it into y*0/y1*0 the 0's can be seen to cancel and then you get y/y1 for any values y,y1 and therefore can take on any of any infinite values.
***** Actually, I think that algorithm doesn't quite simulate a division by zero because, for any value you insert as a divisor (if you swapped "int(n) - 0" for ,say, "int(n) - 3", for example), you'd still have an infinite loop (because the condition for the while loop will always be true and there is no condition for it to actually stop).
A true general algorithm for a division of integers would be something like that:
-----------------------------------------------------------------------------------------------
n = int(raw_input("Insert the dividend: "))
m = int(raw_input("Insert the divisor: "))
c = 0
result = n
while True:
result -= m
if result < 0:
remainder = result + m
break
c += 1
print c
print "%d/%d = %d with a remainder of %d"%(n,m,c,remainder)
--------------------------------------------------------------------------------------------------
If you insert 0 as the divisor, the "c" values will explode into infinity on the screen until you hit that close button, however, inserting other positive integer values would return normal division results. :D
(also written in Python, because screw it, i'm on that lazy train too \o/)
Robert
Dividng is reverse of multiplyin so:
4/3 is
4 * (1/3)
and a proof of this is that:
(4/3) * 3 = 4
(a/b) * b = a
so:
if b = 0 and a is any N then a =/= a which as answer is not in set N because any a in this set is equal to itself
(4/0) * 0 = 0 ==> 4=0
The result of this nonsene came from the set. Any result on the corpus of N must result in the corpus of N and 0 is not in even in the set of N.
Szloma Josif Point being?
Robert Fennis
Set dosent include result.
Need a larger set with algebra over biger corpus with a diferent ring and more dimensions.
And of course problem is solved, directX is working perfectly without gimbal lock on this wee issue of dividing by zero.
Mathematician: "you can't divide by zero"
Engineer: "Just watch me!!!"
That must be what Denny Pate, the designer of the FIU bridge did, and then the Boeing engineers that designed the 737MAX followed his lead.
Oh c'mon, it's usually the otherway round. Unlike mathematicians, engineers are too boarged down with deadlines and budget constraints that they hardly have any luxury to play with theories and concept. Otherwise the boss would show them the door 😅
Donald Knuth disagrees about 0° bring undefined. He talks about that in a very interesting article named "Two Notes on Notation". I recommend reading it, it has convinced me that 0°=1 is the best choice.
I agree with him and I point out that arguments about limits have at least 3 problems.
1. An operation is nothing but a function and a function has no obligation to be continuous so the possible value of any limits with (x, y) going to (0,0) don't need to have any relation with the value of such operations at (0,0).
2. Operations, like any function, don't have the obligation of being continuous. Add to this the previous point.
3. Before you prove anything in mathematics you must first have clear definitions (or axioms) of everything you are dealing with. So in order to prove anything (like limit values or inexistence of limits) about the cited operations you must FIRST define them entirely and that means also define their domains and values at any point, including possibly (0,0). So formally speaking you must decide about 0° (and all x^y values) BEFORE saying anything about limits involving x^y.
That said, since 0 is a natural number, we should define operations involving natural x and 0 in the context of natural numbers before passing to limits. And in the particular case of 0° there are many uses of the identity 0°=1 in discrete contexts and that makes many people decide for adopting it.
I always hated math and no one I know likes it, but it is nice to see people with a real passion and love for maths/numbers to make it more interesting
I divided by zero and my calculator transformed into Optimus Prime and rolled out
my calculator prints out the entirety of Romeo & Julie and then splits into several people and performs it.
@@bsm239 XD
So lame
Greetings Mortal So lame
"And people will yell at me if i say its infinitely different" i lost it
these videos are really interesting, great channel!!!
0 *exists*
Numberphile: It's free real estate.
2:20 - 2:30 But 1/0 is equal to blue.
If I had numberphile as my math professor, Vsauce as science, I would top the school.
I don't think Vsauce knows curriculum science. He is only interested in the abstract and ambiguous topics of science.
Matt: you can’t divide by 0
Little kid: *gasps*
James: *draws two circles to mean infinity*
Me: *gasps*
Thanks !Happy new year!
For some reason, the way Matt writes his "x" is deeply unsettling to me.
Shouldn't be allowed to work as a mathematician if you write x like he does LOL
How else would you do it? Like a normal written x?
@@clarkeysam yes, you cross two lines like a respectable human being.
Tauno Kekkonen no, that x is anti calculus.
You want a nice, curvy and sexy x.
He writes it just like how you learn to write your x in cursive in second grade
0^0=Spaghetti
Hm I thought it was banana
+Bearboy03
No, 0^Spaghetti is Banana. You were close, though. :)
+Naveek Darkroom 0^0 = 0
+Marcus Johnson Lol - Unless you are having fun, 0 ^ 0 is not 0 :P it's.............................1
+Marcus Johnson - You sound like a guy back in the 17th Century, "0 is nothing"??????????? Just...wow... I laughed quite a lot when i read that... You made my day XD
5:53 From a programming perspective - it's almost always going to be hardcoded on a divide function to throw an error if the divisor is zero. Having a step counter (detecting the "exploding" in a direction) on an iterative process is so prone to fatal mistakes that most people wouldn't code it that way as the sole way to catch a common problem.
That said, there is a third option. You can have your iteration record its result and then have the next iteration compare against that result. Even if it's not truly a non-terminating iteration, it WOULD be one if those match because it would mean the precision of your value prevents you from getting anything out of further iterations. (Identical input in a deterministic function gets the identical output every time.) This is a more robust way of quickly finding the limit of your calculation ability on a given problem and it happens to catch "divide by zero" while it's at it.
There's also the 4th check, verifying if the precision limit has been reached. Most code you see now will use a large signed integer to store values while it's calculating while displaying FAR less. For example, the Windows calculator has 36 characters on its output screen so they're likely using a 64 bit "double precision" value to store digits. This is a standard, useful so you don't hit errors with your custom "38 bit double" but you can decide within your loop if you've calculated far enough that the user won't see it.
Frankly, what most programmers would do is all 4.
- The divide function will generally already be built into the code for any compiler with the "divide by zero" error in place, but if you're coding at a more basic level you would do that yourself. You always call the same divide function for the actual division of any 2 numbers, so the 0 is always caught even if it shows up later in the process.
- The iteration loop is given an array for the previous iterations results. At the end of the iteration it calls a separate function to iteratively compare the latest result against the array. The comparing function returns a value and the loop uses that value to determine if it needs to stop.- Any time an iterative process kicks off, you give it a number of iterations it's allowed and the loop can either give an error or a partial answer once the iteration limit is reached. You can also make this smarter with better calculators so a complex loop is given fewer iterations or the whole process is given a set number of iterations to allocate.
- The iteration loop is given an array for the previous iterations results. At the end of the iteration it calls a separate function to iteratively compare the latest result against the array. The comparing function returns a value and the loop uses that value to determine if it needs to stop.
- Finally check if the value change is of low enough precision to matter.
Essentially, you compartmentalize your functions so that they catch the sorts of problems they each introduce. The top level, division, fast-ejects known problems like 0 so you're not burning through limited resources on a calculator powered by a cheap solar cell or a watch battery (simple calculators) or taking a long time deep in an operation of a more complex calculator. The loop counter then catches ANY iterative process that takes too long, as you mentioned and it calls a halt to the loop, either giving out an approximation if it can, or an error if it can't. Finally, inside the loop it has a faster way to catch repeating processes, whether that's something like 1/3 where it repeats fast, or 1/81 where it takes a while to repeat. Finally, it figures out where it can stop caring, so something like 1/9973, it would loop through 30+ times, realize it hit its precision limit, and return the bits it has so far rather than continuing out 9966 times and realizing from the remainder and last calculated value that it's repeating.
That was beautifully explained.
11:50 I see what you did there, NAUGHTy
Freaky Fred :)))))
Who said that mathematicians can't be naughty?
If 1/0 equals infinity, the that means infinity times 0 equals 1, which is not possible.
0χ(+-infinity) is an undefined number so it can be 1 as well
You can't just treat infinity as any other number. It is an idea, not a value you can do operations on.
I did not say that infinity is any number.I said 0x(infinity) is an undefined number so you dont know whether it is equal to 1 or not , therefore you can not say that 0x(infinity)=1 is an impossible equation, it is an undefined equation. And you can do operations on infinity. For example: (1/infinity)=0 and: (+infinity)(-infinity)=-infinity
+Tomas Cena I was refering to the main comment :)
DrDerp42
This is a dangerous subject
The intuitive way I think about division by zero not being infinity: treat division like the number of subtractions from a number to get to zero (like in the video). Keep in mind that there are infinite processes that give sensible answers, like Sum of 1/2^n from 0 to infinity equals 2. However, even if you take away zero from a number an infinite number of times you still wouldn't get to zero. Therefore, there can't be an answer to a number divided by zero. Similarly, 0/0 can be interpreted as any number for the same reason.
I noticed this many times that 0/0 tends to be anything in different cases but today i understood its real reason
6:14 the answer for the calculator is defined by IEEE floating point standards and generally requires that software implement exception handling such that when the processors encounters a divide instruction with a zero operand it generates the divided-by-zero exception so the software can decide what to do.
anything divided by zero = 42
come on people! don't you know your DNA?
You must be descended from the telephone sanitizer;)
so zero = life?
ATBPjako Since 0 can be defined as Nothing or None, 0 = 42 could mean "You have no life"
If anything divided by 0 = 42 and 42 is the answer to the ultimate question, then anything divided by 0 (x\0) IS the ultimate question. That means that the answer to x\0 is the meaning of life and everything else. Come to think of it, lim x->0 = infinity (positive or negative) but never reaches 0 itself - it's composed of everything in the universe except for a point where there is nothing. Oh man, I don't know, if you catch my drift. I'll call it Caldoon-Adams-Julekmeister's Law of Relative Existence :)
Even the number of likes on your post has been divided by zero
CompSci here: there is a standard called IEEE which defines mathematics for computers
it says that for any positive, non-zero 'x': x/0 = inf, -x/0 = -inf
and that 0/0 = nan
where inf is not a number, just a representation of infinity (and it also defines operations such as inf + x = inf and so on...)
and nan is just 'not a number'
I must add that this is just the standard most programming languages uses for mathematical operations, not how it works in actual mathematics
computers have hardware limits where mathematics doesn't
CONGRATULATIONS MATT ON
1M SUBSCRIBERS!
I consider 0/0 to be a feature, not a bug.
Simple Algebraic Rearrangement tells us:
If Anything * 0 = 0,
Then 0 / 0 = Anything.
Intuitively, 0 / 0 represents the question "what number can you multiply by 0 to get 0", to which the answer is clearly "Anything".
If 0/0=1 and 0/0=2 then 1=2 by the transitive property. If you allow that, the you can say that every number is equal to each other.
The transitive property doesn't always work that way.
√1 = 1 and √1 = -1, but 1 ≠ -1
MumboJ the square root symbol like that that is a function. when you use the square root symbol like that you are taking the principle root which is always positive. You mean to right x^2=1 which has to possible solutions that make that statement true. If you want the negative value you have to put the negative sign in front.
Which is exactly how 0/0 works.
It is a rearrangement of 0x=0, to which anything is a solution.
Function Symbols are often used to represent this concept, and the phrasing I used was not incorrect.
MumboJ a square root of a number can be either positive or negative. Its because both positive squared and negative squared are positive. Example: √1=±1
√4=±2
√2≈±1.414
But if you push too hard...even numbers got LIMITS
that because they set a limit in the first place, they set it to zero, if i don't set any limit, it would go forever
Salvador Allende those limits are "errors" (paradoxes) in the system we made
He's quoting a Mos Def song called Mathematics.
If you don't know, now you know...
What about odd numbers, they got limits too?
Zero: Oh you’re approaching me?
I love how old this video is in terms of how different Matt Parker is
I met this channel a while ago, when i was in highschool and used to watch every video. Now, as i'm graduating in mathematics i come back and rewatch the same videos, but now in a different perspective. Numberphile was one of the main reasons i decided to study math in college, despite all flaws.
From the software engineering perspective I'd say that I highly doubt that any commonly used calculator uses iterative process to get an answer for X/0. It's just a check in the code: if operation is division and second argument is 0 then print "Error". So, the first guess is much closer to reality
Definitely more likely. Calculators only really do addition and subtraction so if you tried to divide by zero it would keep subtracting by zero an infinite amount of times, just like they demonstrated in the video. Its gotta be programmed to check for a non zero number to keep it from entering an infinite loop, that seems like the best solution
This is how most applications work, most of the time its baked into the compiler (Roslyn)
It's normally an exception, stopping your execution. If you have a divide by zero in your equation and you don't stop, you're in la-la land. CPUs handle integer division (which give division by zero and overflow), languages have standard libraries for floating point. The standard is to have zero, NaN (Not a number), Inf, and -Inf as distinct results. Most calculators now have this as well (processors are very cheap).
NaN is different than infinity. Infinity normally means it encountered a number which exceeded the maximum value (e.g. 300 factorial), and infinity times zero is zero. Any math operation using NaN gives you NaN as a result. You also can have exceptions for overflow/infinity, and there may be cases where you want to know when you underflow (if you have x and y, which are not zero, but you get zero because the number is too small.. that's not normally one you worry about).
A difficult problem in programming is when you have a one-off problem, where it goes into la-la land, and takes a few steps before it dies. Math is one of those things.
Unless someone forgets to tell the computer not to attempt the subtraction, in which case the computer may crash, which happened to an American warship, computers down for a day
Happy New year sir
In the early 70's the first electronic calculator I got was a Casio, and if you tried to divide any number by 0, it would display all 9's (don't remember how many) and start counting down from that huge number! Entertaining...
An interesting thing r.e. this divide by 0, is how you can handle it on the HP 49 and 50 calculators. They have a flag for divide by zero; you can set it to either produce an error or to result in the largest number the calculator can handle.
Drawing X as two half circles?
Classic Parker Square.
Lots of people do this. You pick up handwriting habits like this so you can more easily distinguish between symbols that look the same. x and the symbol for cross products, for example, look similar and will confuse people unless you draw the letter x as half circles.
This is a relatively common convention tbh. It was specifically adopted so that 'x' would be more readily distinguishable from the multiplication symbol in mathematical proofs and textbooks. A common alternative was to use * as the multiplication symbol, as most scientific calculators do.
That looks more like a khi
@@Bignic2008 thats actually pretty cool. Definetivly reasonable. 👍
@@smockboy makes sense and is smart 👍
1/0=Blue... i'm going to write that next time i write a calculator program..
In my grade 12 history class I did a paper about the origins of zero and the teacher had to get a math teacher to grade it.
I got a 90% because my writing is dry and transactional like an instruction manual and apparently that counts.
"it starts to fall apart when you go to the complex plain" perfectly sums up my brain at this point in the video.
I divided by 0 and my paper set on fire
Yea, I tried it on my Samsung 8 and it exploded
BEANS
No that is what 2 beans add 2 beans is
That is actually some beans
Or a really small casserole
If there were a god i could believe in, it would be a lot like *0*. It is, after all, the sum of all the numbers. As the sum of all things it lacks nothing. It is all the points of a circle taken together, defining them all, and yet not a part of that circle. *0* is present anywhere in _infinite_ abundance and yet is hidden. It is the goal of countless equations - although only if you feel like arranging them that way. And by giving them a start, it gives numbers their uniqueness and potential, without ever dividing them. The beginning of all things and the end of all things. A connection between the past and the future, left and right, top and bottom, When we connect (in all sorts of ways) we feel love (of all kinds), So the ultimate connection point, 0, is love.
Well, if there were such a God, it would have my deepest respect - to be all this, to do all this, without even existing... ;)
This is incredibly deep...
But zero exists... doesn't it?
oh_no_mrbill But 0 is nothing..so does it? ;)
vovka-morkovka It was meant to be a joke.
"I guess, whether or not something exists is up to whether or enough of us agree that it exists."
Well thats not very mathematical, because wether or not something exists in math is either proofable OR defined. and you can define anything.
And for Maths it doesn't matter if it exists in reallity
Imagine a world without the existence of zero in maths. No way to explain absolute values, no divider separating positive and negative integers. A blank unknown dividing them would be interesting. Haha
I laughed when you said -- when dealing with the different values towards which 1/x tends when you start with positive vs negative one -- that maybe the value wraps around the entire world, such that positive infinity connects to negative infinity, because that is exactly how my high-school teacher explained tangents.
2:10, thats one way of drawing an 'x'!
Common in math, I've heard.
im surprised they didn't explode
I think James' first description is pretty much perfect. You just keep subtracting a number until you get to zero. If you did 20 - 0 an infinite number of times, you'd still end up with 20, because every step leaves you with 20. So infinity essentially has no effect on subtracting (or dividing by) zero.
"Sometimes, when you divide by 0, you could get 5." - My calc teacher.
Five likes too
Matt: You cannot divide by zero!
Ring with nonzero zero divisors: Hold my beer...
OH GOD NO... He's gonna divide by zer....................
+Sarge!!! BOOOM
Dewey russian kh/ch -Хх
hey i heard someone divided by zero. is everyone okay?
(Bomb explodes)
⚫️
Dividing by zero is such a Parker Square move.
lol, only numberphile fans will get this.
There's something so wierdly soothing about the way matt writes X's
Thank you very much for the pleasure, I love math, it is such a beauty!